Question: Is ${203758}$ divisible by $3$ ?
Solution: A number is divisible by $3$ if the sum of its digits is divisible by $3$ . [ Why? First, we can break the number up by place value: $ \begin{eqnarray} {203758}= &&{2}\cdot100000+ \\&&{0}\cdot10000+ \\&&{3}\cdot1000+ \\&&{7}\cdot100+ \\&&{5}\cdot10+ \\&&{8}\cdot1 \end{eqnarray} $ Next, we can rewrite each of the place values as $1$ plus a bunch of $9$ s: $ \begin{eqnarray} {203758}= &&{2}(99999+1)+ \\&&{0}(9999+1)+ \\&&{3}(999+1)+ \\&&{7}(99+1)+ \\&&{5}(9+1)+ \\&&{8} \end{eqnarray} $ Now if we distribute and rearrange, we get this: $ \begin{eqnarray} {203758}= &&\gray{2\cdot99999}+ \\&&\gray{0\cdot9999}+ \\&&\gray{3\cdot999}+ \\&&\gray{7\cdot99}+ \\&&\gray{5\cdot9}+ \\&& {2}+{0}+{3}+{7}+{5}+{8} \end{eqnarray} $ Any number consisting only of $9$ s is a multiple of $3$ , so the first five terms must all be multiples of $3$ That means that to figure out whether the original number is divisible by $3 $ , all we need to do is add up the digits and see if the sum is divisible by $3$ . In other words, ${203758}$ is divisible by $3$ if ${ 2}+{0}+{3}+{7}+{5}+{8}$ is divisible by $3$ Add the digits of ${203758}$ $ {2}+{0}+{3}+{7}+{5}+{8} = {25} $ If ${25}$ is divisible by $3$ , then ${203758}$ must also be divisible by $3$ ${25}$ is not divisible by $3$, therefore ${203758}$ must not be divisible by $3$.